Problem 41 - Introductory Algebra for College Students

Question

Factor completely, or state that the polynomial is prime. $$-3 x^{2}+75$$

Step-by-Step Solution

Answer

The factorized form of \(-3x^2 +75\) is \(-3(x - 5)(x + 5)\)

Step 1 - Check if it's a difference of squares

Looking at the expression \(-3x^2 + 75\), arrange it so the result resembles a difference of squares. It becomes \(-3(x^2 - 25)\). We see that 25 is a perfect square, so it's indeed a difference of squares.

Step 2- Factorize further

Factorize the difference of squares from Step 1, using the fact that \(x^2 - a^2\) can be factored as \((x - a)(x + a)\). We have, \(-3(x^2 - 25) = -3(x - 5)(x + 5)\).

Key Concepts

Difference of Squares

The difference of squares is a key concept when factoring polynomials. It involves two terms where each term is a perfect square, and they are separated by a subtraction sign. The general form is: \[a^2 - b^2\] This can be factored into: \[(a-b)(a+b)\] This is a powerful tool for simplifying expressions like \(-3(x^2 - 25)\).

• The expression \(x^2\) and \(25\) are perfect squares.
• \(25\) is the square of 5, making both terms perfect squares.

Therefore, \(x^2 - 25\) can be rewritten as \((x - 5)(x + 5)\). This makes it straightforward to factor the polynomial by noticing it as a difference of squares.

Perfect Squares

Perfect squares are expressions that result from squaring a polynomial term. They are essential when identifying factors of polynomials. Examples include numbers like 25, which is \(5^2\), or algebraic terms like \(x^2\).

• Perfect squares often help in recognizing patterns that allow us to apply simple factoring techniques.
• In our example \(-3x^2 + 75\), \(x^2\) and \(25\) are perfect squares which can simplify factoring.

Recognizing these squares allows us to leverage the difference of squares formula for easier factorizations.

Algebraic Expressions

Algebraic expressions are mixtures of numbers, variables, and operations. They form the basis of equations and can be manipulated by operations like addition, subtraction, multiplication, and division. Factorization is a critical skill for simplifying complex algebraic expressions. It helps in solving equations and simplifies expressions for further operations.

• Understanding the structure helps in applying the correct factoring method, such as grouping or recognizing common patterns.
• In \(-3x^2 + 75\), recognizing the product of a constant and a perfect square expression allows us to factor efficiently.

The ability to decode an algebraic expression simplifies the process of finding solutions or simplifying further calculations, making math problems more approachable.